Related papers: The Fujimoto Conjecture via Total Positivity

In this paper, we show that if $m$ and $n$ are distinct positive integers and $x$ is a nonzero real number with $\Phi_m(x)=\Phi_n(x)$, then $\frac{1}{2}<|x|<2$ except when $\{m,n\}=\{2,6\}$ and $x=2$. We also observe that 2 appears to be…

Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1\geq \mu_2\geq \cdots\geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma=\sigma(G)$ $(1\leq \sigma\leq n)$ be the largest positive integer such…

Let $p$ be an odd prime, $F/{\Bbb Q}$ an abelian totally real number field, $F_\infty/F$ its cyclotomic ${\Bbb Z}_p$-extension, $G_\infty = Gal (F_\infty / {\Bbb Q}),$ ${\Bbb A} = {\Bbb Z}_p [[G_\infty]].$ We give an explicit description of…

Let $\mathbb{F}$ be a field, and $n \geq r>0$ be integers, with $r$ even. Denote by $\mathrm{A}_n(\mathbb{F})$ the space of all $n$-by-$n$ alternating matrices with entries in $\mathbb{F}$. We consider the problem of determining the…

Given $d$-dimensional standard Gaussian vectors $\boldsymbol{x}_1,\dots, \boldsymbol{x}_n$, we consider the set of all empirical distributions of its $m$-dimensional projections, for $m$ a fixed constant. Diaconis and Freedman (1984) proved…

A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap…

In 1992, Erd$\H{o}$s and Hegyv$\'{a}$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, Fang and Chen showed that for any positive integer m, there exists infinitely…

Zeckendorf proved that any integer can be decomposed uniquely as a sum of non-adjacent Fibonacci numbers, $F_n$. Using continued fractions, Lekkerkerker proved the average number of summands of an $m \in [F_n, F_{n+1})$ is essentially…

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees…

We collect a number of striking recent results in a study of dimers on infinite regular bipartite lattices and also on regular bipartite graphs. We clearly separate rigorously proven results from conjectures. A primary goal is to show…

Alphatrion conjectured that it is possible to label the vertices of an $n$-dimensional hypercube with distinct positive integers such that for every Hamiltonian path $a_1, \dots, a_{2^n},$ we have $a_i + a_{i+1}$ prime for all $i.$ We prove…

Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erd\H{o}s and Szekeres proved that f(n) exists and…

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in…

For any $m = 3 \left( 2n + 1 \right) with \ n \in \mathbb{N^*} ,$ the prime counting function $\pi(m) = 4 + \left \vert A_4(m) \right \vert + 2 \left \vert A_6(m) \right \vert $ where $A_6(m) $ and $ A_4(m) $ are the sets of Twin Primes and…

We classify all solution triples with Fibonacci components to the equation $a^2+b^2+c^2=3abc+m,$ for positive $m$. We show that for $m=2$ they are precisely $(1,F(b),F(b+2))$, with even $b$; for $m=21$, there exist exactly two Fibonacci…

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in…

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most…

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…

In 1948, Erd\"{o}s and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that…

The projective algebra p(M;F) (i.e the collection of all projective vector fields)of a Finsler space (M;F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket. The projective algebra of Einstein metrics has been…